Heat 2 teaspoons of the oil in a nonstick skillet over medium heat.Add the onions and saute until translucent and beginning to brown, about 4 minutes. Add the garlic and continue to cook for another 1 minute. Transfer the onion mixture to a medium bowl and allow to cool.Heat remaining 2 teaspoons of oil in the skillet and saute the mushrooms for about 6 to 7 minutes. Make sure they are thoroughly cooked. Remove from heat and cool completely.Spray an oval baking dish with cooking spray. (or generously cover in butter)Arrange the bread cubes in the dish.In a large bowl, beat the eggs, milk and mustard until incorporated.Add mushrooms, onion-garlic mixture, broccoli, parmesan and cheddar cheeses, sun-dried tomatoes (drained), and salt and pepper and stir to incorporate.Pour mixture over bread, making sure liquid saturates bread. Cover with plastic wrap and refrigerate overnight, or at least 8 hours.Preheat the oven to 350 degrees F. Remove the plastic wrap from strata and bake for 55 to 60 minutes, or until top forms a light brown crust.